can anybody help me mathematically? I am searching a proof why finite difference methods are not necessarily conservative but finite volume methods are. This is for me the main advantage of FV-schemes....

Some finite difference methods can also be re-written as a finite volume method. See Godlweski and Raviart, "Hyperbolic systems of conservation laws" for a mathematical condition which makes a scheme "finite volume".

I think you misunderstand my point. Plus, conservation and accuracy are in this discussion separate issues.

A FV scheme is written in integral form. Thus for every single cell, the sum of the fluxes across the cell faces equals the time rate of change for the cell state. Always. In this manner, the mathematical statement of conservation is strictly enforced, irrespective of cell size or averaging.

Yes, accuracy depends on cell size and extrapolation from the cell center to the faces to evaluate higher order fluxes, but that is a separate matter.

In a FD scheme, the mathematical statement of conservation is weakly enforced, and only holds in the limit of grid spacing going to zero.

"A FV scheme is written in integral form. Thus for every single cell, the sum of the fluxes across the cell faces equals the time rate of change for the cell state. Always. In this manner, the mathematical statement of conservation is strictly enforced, irrespective of cell size or averaging."

>> Ok, I think to understand what you mean.

"In a FD scheme, the mathematical statement of conservation is weakly enforced, and only holds in the limit of grid spacing going to zero."

>> So, can you imagine a concrete problem where FD-formulation would lead to a problem (=oscillation or divergence) where FV scheme still produces a senseful result??? It would help me a lot to see that with a concrete mathematical problem....

It is not really a matter of oscillation or divergence, those are typically stability issues.

For an internal flow, just think of flow through a pipe. The old finite difference based codes had trouble making the massflow at the end of the pipe equal the massflow at the beginning of the pipe. The conservation of mass was only weakly enforced. If you threw enough grid at it you could reduce the mass error. The more modern FV codes have a much easier time conserving mass.

+ So, first of all we can conclude that FD have no "natural" algorithm to proove conservation - in contrast to FV.

The case which was announced by Joe means that with FD (rho*c)_(i+1) is different than (rho*c)_(i-1) - when we use a big Dx; like three nodes....But in this case (rho*c)_(i-1) is kind of a boundary condition. So we know that exact solution and with an explicit scheme we can directly compute the next value (rho*c)_(i+1). Where is the source for the error? Why does it only happen with the FD-formulation??????

If the area is changing, why do you not include that in the finite difference approximation as well? What you are looking for is termed the "telescoping" property of finite difference schemes, and the classic text by P. Roache contains a good description. A finite difference scheme can be just as conservative as a finite volume scheme, but requires attention to the construction of the terms. Marcel Vinokur wrote a good treatise on this a number of years ago - unfortunately the title of the paper escapes me at the moment. Try googling for Vinokur. As noted by Praveen, it is possible to construct a conservative scheme using finite differences within finite volume framework.

The integral form is not required to generate your eq. 2. The PDE is differenced in a finite-volume sense by writing the flux derivative (e.g. dE/d(xi)) as a difference of fluxes at the differential cell faces, E(i+1/2) - E(i-1/2). This is a finite difference formulation that possesses a flux telescoping property as long as certain rules are followed in constructing the face fluxes. Thus, it can represent a conservative finite difference formulation.

In general discontinuous problems do need to be handled using the integral (weak) formulations. Fortunately, by applying certain entropy conditions we can also handle discontinuous flow fields using the differential form of the equations - the entropy conditions ensure that we get the correct weak solution. I suggest that you grab a copy of Leveque's "Numerical Methods for Conservation Laws".

The differential equations can be written in a divergence form that allows us to generate the correct solution by discretizing the equations using finite difference approximations. I'm not sure what you mean by "no divergence term". The ability to cast the equations into a divergence form is what allows us to move from an integral form to a differential form.